Question : Are all polygons drawn as a combination of the triangles?

Description:

I am drawing a surface where I am given the points on a plane and their colormap values. I have a predefined 1D texture using which and the colormap values of the points I draw the surface.

I am drawing the surface as a combinations of triangles.

Here is the problem:

In a surface with all the points colormap values same(say 0) except one point’s colormap value(lets say its values is 1), the surface drawn has a square or a rhombus of the different color(for the different colormap value point) depending on the position of the point. The rhombus is half the size as the square.

Now this is happening because when a surface is drawn as a combination of triangles, each point is either a part of 8 triangles or 4 triangles depending on its position. Due to this, the texture mapping effects either 4 triangles or 8 triangles around the point giving me a square or a rhombus.

So I thought that instead of drawing the surface as combination of triangles, I should draw the surface as a combination of squares(QUADS) and this should solve the problem. I tried this, but the same behavior is seen in this also.

This led me to conclude that these squares are being drawn as two triangles internally or the texturing is occurring for three points at once. Am I right about this? And how can this be fixed?